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We show the existence of a {\\em smallest} invariant set (with respect to inclusion) for $\\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\\x) = r_i \\boldmath{x} + b_i$ on $X=\\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. 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